Lusternik-schnirelmann category of spin(9)

First we give an upper bound of cat(E), the L-S category of a principal G-bundle E for a connected compact group G with a characteristic map α:∑V→G. Assume that there is a cone-decomposition {Fi | 0 ≤ i ≤ m} of G in the sense of Ganea that is compatible with multiplication. Then we have cat (E)≤Max(m+n, m+2) for n ≥ 1, if α is compressible into F_n ⊆ F_m ≃ G with trivial higher Hopf invariant H_n(α). Second, we introduce a new computable lower bound, Mwgt(X; F2) for cat(X). The two new estimates imply cat(Spin(9)) = Mwgt(Spin(9); F2) = 8 > 6=wgt(Spin(9); F2), where (wgt-;R) is a category weight due to Rudyak and Strom.